Simplify 3 square root of -18+5 square root of -12
The problem is asking to perform arithmetic operations with complex numbers that are represented in the form of square roots of negative numbers. Specifically, it involves simplifying an expression that combines two terms, each of which is a real number multiplied by the square root of a negative number. This involves using the properties of imaginary numbers and simplifying the expression by combining like terms and possibly performing operations with complex numbers.
$3 \sqrt{- 18} + 5 \sqrt{- 12}$
Answer
Solution:
Simplification Process
Step 1:
Express $-18$ as $-1 \times 18$. Thus, we have $3 \sqrt{-1 \times 18} + 5 \sqrt{-12}$.
Step 2:
Separate the square root of the product into the product of square roots: $\sqrt{-1 \times 18} = \sqrt{-1} \cdot \sqrt{18}$. Now, the expression is $3 (\sqrt{-1} \cdot \sqrt{18}) + 5 \sqrt{-12}$.
Step 3:
Replace $\sqrt{-1}$ with the imaginary unit $i$. The expression becomes $3 (i \cdot \sqrt{18}) + 5 \sqrt{-12}$.
Step 4:
Decompose 18 into its prime factors, $18 = 3^2 \times 2$.
Step 4.1:
Extract the square root of 9 from $\sqrt{18}$: $3 (i \cdot \sqrt{9 \times 2}) + 5 \sqrt{-12}$.
Step 4.2:
Express 9 as $3^2$: $3 (i \cdot \sqrt{3^2 \times 2}) + 5 \sqrt{-12}$.
Step 5:
Remove the perfect square from under the radical: $3 (i \cdot (3 \sqrt{2})) + 5 \sqrt{-12}$.
Step 6:
Rearrange the terms to place the constant before $i$: $3 (3 \cdot i \sqrt{2}) + 5 \sqrt{-12}$.
Step 7:
Multiply the constants outside the radical: $9 (i \sqrt{2}) + 5 \sqrt{-12}$.
Step 8:
Express $-12$ as $-1 \times 12$: $9 i \sqrt{2} + 5 \sqrt{-1 \times 12}$.
Step 9:
Separate the square root of the product into the product of square roots: $\sqrt{-1 \times 12} = \sqrt{-1} \cdot \sqrt{12}$. The expression is now $9 i \sqrt{2} + 5 (\sqrt{-1} \cdot \sqrt{12})$.
Step 10:
Replace $\sqrt{-1}$ with $i$: $9 i \sqrt{2} + 5 (i \cdot \sqrt{12})$.
Step 11:
Decompose 12 into its prime factors, $12 = 2^2 \times 3$.
Step 11.1:
Extract the square root of 4 from $\sqrt{12}$: $9 i \sqrt{2} + 5 (i \cdot \sqrt{4 \times 3})$.
Step 11.2:
Express 4 as $2^2$: $9 i \sqrt{2} + 5 (i \cdot \sqrt{2^2 \times 3})$.
Step 12:
Remove the perfect square from under the radical: $9 i \sqrt{2} + 5 (i \cdot (2 \sqrt{3}))$.
Step 13:
Rearrange the terms to place the constant before $i$: $9 i \sqrt{2} + 5 (2 \cdot i \sqrt{3})$.
Step 14:
Multiply the constants outside the radical: $9 i \sqrt{2} + 10 i \sqrt{3}$.
Knowledge Notes:
The problem involves simplifying a complex expression with square roots of negative numbers. The key knowledge points are:
Imaginary Unit ($i$): The imaginary unit $i$ is defined as $\sqrt{-1}$. It is used to express the square root of negative numbers in terms of real numbers.
Square Root Properties: The square root of a product $\sqrt{ab}$ can be expressed as the product of square roots $\sqrt{a} \cdot \sqrt{b}$.
Simplifying Square Roots: When simplifying square roots, any perfect square factors can be taken out from under the radical. For example, $\sqrt{a^2 \cdot b} = a \sqrt{b}$ if $a$ is a real number.
Combining Like Terms: When simplifying expressions with imaginary numbers, combine like terms by adding or subtracting the coefficients of the terms with the same imaginary part.
Factorization: Breaking down composite numbers into their prime factors can help simplify the square roots of those numbers.
Radical Manipulation: The manipulation of radicals involves extracting and simplifying terms under the radical sign to make the expression more straightforward.
By applying these principles, the original expression is simplified to an expression involving imaginary numbers and simplified radicals.
